On Estimates for Norms of Some Integral Operators with Oinarov's Kernel

Abstract

In this work, we give estimates for the norm of the integral operator

\( H : L_{p,v} \to L_{q,u}, \quad (Hf)(x) := \int_a^x k(x,t)f(t) \, dt \) \quad (0.1)

with the so-called Oinarov's kernel \(k(x,t)\) in the weighted Lebesgue spaces

\( L_{p,v} = \left\{ f : \|f\|_{p,v}^p := \int_a^b |f(t)|^p v(t) \, dt < \infty \right\} \)

and

\( L_{q,u} = \left\{ f : \|f\|_{q,u}^q := \int_a^b |f(t)|^q u(t) \, dt < \infty \right\} \),

in the case \(1 < q < p < \infty\).